# Liquid mirror telescope

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A liquid mirror telescope

Liquid mirror telescopes are telescopes with mirrors made with a reflective liquid. The most common liquid used is mercury, but other liquids will work as well (for example, low melting alloys of gallium). The container for the liquid is rotating so that the liquid assumes a paraboloidal shape. A paraboloidal shape is precisely the shape needed for the primary mirror of a telescope. The rotating liquid assumes the paraboloidal shape regardless of the container's shape. To reduce the amount of liquid metal needed, and thus weight, a rotating mercury mirror uses a container that is as close to the necessary parabolic shape as possible. Liquid mirrors can be a low cost alternative to conventional large telescopes. Compared to a solid glass mirror that must be cast, ground, and polished, a rotating liquid metal mirror is much less expensive to manufacture.

Isaac Newton noted that the free surface of a rotating liquid forms a circular paraboloid and can therefore be used as a telescope, but he could not actually build one because he had no way to stabilize the speed of rotation.[citation needed] The concept was further developed by Ernesto Capocci of the Naples Observatory (1850), but it was not until 1872 that Henry Skey of Dunedin, New Zealand constructed the first working laboratory liquid mirror telescope.

Another difficulty is that a liquid metal mirror can only be used in zenith telescopes, i.e., that look straight up, so it is not suitable for investigations where the telescope must remain pointing at the same location of inertial space (a possible exception to this rule may exist for a mercury mirror space telescope, where the effect of Earth's gravity is replaced by artificial gravity, perhaps by rotating the telescope on a very long tether, or propelling it gently forward with rockets). Only a telescope located at the North Pole or South Pole would offer a relatively static view of the sky, although the freezing point of mercury and the remoteness of the location would need to be considered. A very large telescope already exists at the South Pole, but the North Pole is located in the Arctic Ocean.

Currently, the mercury mirror of the Large Zenith Telescope in Canada is the largest liquid metal mirror in operation. It has a diameter of six meters, and rotates at a rate of about 8.5 revolutions per minute.

## Explanation of the equilibrium

In fluid mechanics, the state when no part of the fluid has motion relative to any other part of the fluid is called 'solid body rotation'. When the liquid has reached a state of solid body rotation, then the dynamic equilibrium can be understood as a balance of two energies: gravitational potential energy, and rotational kinetic energy. When a fluid is in solid body rotation it is the lowest state of energy that is available, because in a state of solid body rotation there is no friction to dissipate any of the energy. In an inertial reference frame, the dynamic equilibrium cannot be understood in terms of an equilibrium of forces. This is because when the liquid is rotating, there is an unbalanced force acting on the liquid – the force of gravity is acting in a vertical direction on the liquid, and the surface of the parabolic dish exerts a normal force on the liquid resting on it. The resultant force is a net centripetal force toward the axis of rotation.

The following discussion is for the case of the liquid as it is rotating in solid body rotation.

The force of gravity (red), the normal force (green), and the resultant centripetal force (blue)

The kinetic energy of a parcel of liquid given by the formula:

$E_{kin} = \frac{1}{2} m v^2$

In the case of circular motion the relation $v = \omega r$ holds ($\omega$ is in radians per second), hence

$E_{kin} = \frac{1}{2} m \omega^2 r^2$

The gravitational potential energy is given by

$E_{pot} = m g h$

where $g$ is the acceleration of gravity and $h$ is the height of the liquid's surface above some arbitrary elevation, for instance, we can set $h=0$ to be the lowest liquid surface.

We set the potential energy equal to the kinetic energy to find the liquid's shape:

$h = \frac{1}{2 g} \omega^2 r^2$

This is of the form $h=kr^2$, where $k$ is a constant, which is, by definition, a paraboloid.

### Alternative method

It is not necessary to invoke the equality of rotational kinetic energy and gravitational potential energy. With reference to the force diagram above, the vertical component of the normal force (green arrow) must equal the weight of the parcel (red arrow), which is $mg$, and the horizontal component of the normal force must equal the centripetal force (blue arrow) that keeps the parcel in circular motion, which is $m \omega^2 r$. Since the green arrow is perpendicular to the surface of the liquid, the slope of the surface must equal the quotient of these forces:

Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of plexiglass. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre.
$\frac{d h}{d r} = \frac{m \omega^2 r}{m g}$

Cancelling the $m$'s, integrating, and setting $h=0$ when $r=0$ leads to

$h = \frac{1}{2 g} \omega^2 r^2$

which is identical to the result obtained by the previous method, and likewise shows that the liquid surface is paraboloidal.

### Rotation speed and focal length

The equation of the paraboloid in terms of its focal length (derived here) can be written as:

$4fh=r^2$

where $f$ is the focal length, and $h$ and $r$ are defined as above.

Dividing this equation by the last one above it eliminates $h$ and $r$ and leads to:

$2 f \omega^2 = g$

which relates the angular velocity of the rotation of the liquid to the focal length of the paraboloid that is produced by the rotation. Note that no other variables are involved. The density of the liquid, for example, has no effect on the focal length of the paraboloid. The units must be consistent, e.g. $f$ may be in metres, $\omega$ in radians per second, and $g$ in metres per second-squared. The angle unit in $\omega$ must be radians.

On the Earth's surface, where $g$ is approximately 9.81 metres per second-squared, this last equation reduces to the approximation:

$f s^2 \approx 447$

where $f$ is the focal length in metres, and $s$ is the rotation speed in revolutions per minute (RPM).

## Liquid mirror telescopes

### Conventional land-based liquid mirror telescopes

These are made of liquid stored in a cylindrical container made of a composite material, such as Kevlar. The cylinder is spun until it reaches a few revolutions per minute. The liquid gradually forms a paraboloid, the shape of a conventional telescopic mirror. The mirror's surface is very precise and small imperfections in the cylinder's shape do not affect it. The amount of mercury used is small, less than a millimeter in thickness.

### Moon-based liquid mirror telescopes

Low temperature ionic liquids (below 130 kelvins) have been proposed[1] as the fluid base for an extremely large diameter spinning liquid mirror telescope to be based on the Earth's moon. Low temperature is advantageous in imaging long wave infrared light which is the form of light (extremely red-shifted) that arrives from the most distant parts of the visible universe. Such a liquid base would be covered by a thin metallic film that forms the reflective surface.

### Space-based ring liquid mirror telescopes

The Rice liquid mirror telescope design is similar to conventional liquid mirror telescopes. It will only work in space; but in orbit, gravity will not distort the mirror's shape into a paraboloid. The design features a liquid stored in a flat-bottomed ring-shaped container with raised interior edges. The central focal area would be rectangular, but a secondary rectangular-parabolic mirror would gather the light to a focal point. Otherwise the optics are similar to other optical telescopes. The light gathering power of a Rice telescope is equivalent to approximately the width times the diameter of the ring, minus a percentage based on optics, superstructure design, etc.

The greatest advantage of a liquid mirror is its small cost, about 1% of a conventional telescope mirror. This cuts down the cost of the entire telescope at least 95%. The University of British Columbia's 6 meter Large Zenith Telescope cost about a fiftieth as much as a conventional telescope with a glass mirror.[2] The greatest disadvantage is, that the mirror can only be pointed straight up. Research is underway to develop telescopes that can be tilted, but currently if a liquid mirror was to tilt out of the zenith, it would lose its shape. Therefore the mirror's view changes as the Earth rotates and objects cannot be physically tracked. An object can be briefly electronically tracked while in the field of view by applying a voltage to the CCD to shift electrons across it at the same speed as the image moves; this tactic is called "drift scanning." Some types of astronomical research are unaffected by these limitations, such as long-term sky surveys and supernova searches. Since the Universe is believed to be isotropic and homogeneous (this is called the Cosmological Principle), the investigation of its structure by cosmologists can also use telescopes which are highly reduced in their direction of view.

Since mercury metal and its vapor are both toxic to humans and animals there remains a problem for its use in any telescope where it may affect its users and others in its area. The less toxic metal gallium may be used instead of mercury but has the disadvantage of high cost. Recently Canadian researchers have proposed the substitution of magnetically deformable liquid mirrors composed of a suspension of iron and silver nanoparticles in ethylene glycol. In addition to low toxicity and relatively low cost, such a mirror would have the advantage of being easily and rapidly deformable using variations of magnetic field strength.[3] [4]

## List of liquid mirror telescopes

Various prototypes exist historically. Following a resurgence of interest in the technology in the 1980s, several projects came to fruition.

• UBC/Laval LMT, 2.65 m, 1992
• NASA-LMT, 3 m, 1995-2002
• LZT, 6 m, 2003-
• ILMT, 4 m, 2011 test